with \(\nu \)  being the random medium fluctuations in (REF ).
Note therefore that in particular the width of \(C\)  is the correlation length of the medium fluctuations.
The Itô-Schrödinger
equation was analyzed for the first time in [1]} and it was derived from first principles
by a multiscale analysis of the wave equation in a random medium in [2]}.
The model (REF ) leads to closed equations for wave field moments of all orders.
We discuss in the appendix the first-order moment equation that is readily solvable.
The second-order one-frequency moment equations are also explicitly solvable, while
the fourth-order equations are not explicitly solvable in the white-noise
paraxial regime, neither in the one-frequency nor in the multi-frequency cases.
However, in a secondary scaling regime
that we denote the scintillation regime we will be able to solve both
the second-order and fourth-order multi-frequency moments.
We will push through this moment analysis in Section .
Before this,
in Section , we discuss the detailed modeling of the two applications which motivates the particular form of the second- and fourth-order
multi-frequency moments that we consider. In Section  we express these moments
in terms of the moments of the Green's function associated with the Itô-Schrödinger equation
(REF ).
